The complete solution of several two-dimensional potential flow problems are reported that deal with the unsteady flow around circular cylinders. Three of the flows considered are induced by an oscillating disturbance near the cylinder. The three elemental disturbances examined are (1) a pulsating source, (2) a pulsating doublet, and (3) a pulsating vortex. The formulas for the force acting on the cylinder due to each of the elemental disturbances were derived by applying the method of images. These results were checked by deriving the equivalent surface distribution of sources to model the cylinder by applying Green’s second identity. The theory helped direct the development of a boundary-integral numerical model described and applied in this paper to solve the unsteady flow around a circular cylinder due to an arbitrarily specified oscillatory disturbance near the cylinder. The numerical method is validated by comparing predictions of the force with the exact solutions. We applied the theory to examine a model problem related to the vortex-shedding force that causes vortex-induced vibration.